Are there review papers, literature reviews in mathematics that describe the recent developments in various fields for a newcomer? Or is the prerequisite knowledge always provided in research monographs and textbooks? That's one can read research papers and make contributions to a field after reading some textbooks or monographs? In physics, all textbooks in my area of interest seem outdated and I should read review articles (see string wiki) that summarize more recent development. I wonder if there are review articles that describe the most important recent achievements in geometry, Analysis, algebra etc.

7 Answers
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In Soviet Union there was special series of "mini-books" surveying recent developments in modern mathematics, top mathematicians Arnold, Kirillov, Manin, Novikov, Sinai, Shafarevich,... wrote volumes of this series. It is really very cool series.
In Russian it was called «Итоги науки и техники» Серия «Современные проблемы математики. Фундаментальные направления», green thin books in hard-covers.
Here is link to content of the series in Russian.

They were translated in English in the series Springer's "Encyclopaedia of Mathematical Sciences".

Let me point out few "student must(enjoy)-read".

First of all is Shafarevich's Basic notions of algebra.
It surveys in accessible level almost all ideas of algebra from groups to K-theory, from rings to algebraic geometry.

S.P. Novikov's Topology, it is analogue of previous one for topology and geometry.

Danilov's Algebraic geometry and schemes - it is covering basic techniques and ideas of algebraic geometry in very friendly way, which I have never met before. Coverage is very wide.

The AMS series, What's Happening in the Mathematical Sciences (AMS link). Now eight volumes. Barry Cipra, a frequent contributor to MO, has authored many of the articles. The
other primary contributor is Dana Mackenzie.

The Best Writing on Mathematics series (Princeton link). Now three years: 2010, 2011, 2012. All edited by Mircea Pitici. These articles tend toward writing for the
general public.

For the last ten years or so, David Eisenbud has organized a "Current Events Bulletin" session at the January joint math meetings, in which, typically, four speakers survey recent work in various fields, with a booklet of the survey papers available at the time of the meeting (and permanently at the website). Here is a portion of Eisenbud's description:

The Current Events Bulletin Session at the Joint Mathematics Meetings,
begun in 2003, is an event where the speakers do not report on their
own work, but survey some of the most interesting current developments
in mathematics, pure and applied. The wonderful tradition of the
Bourbaki Seminar is an inspiration, but we aim for more accessible
treatments and a wider range of subjects.

I. If a newcomer means a well educated person without a special mathematical education
(a physicist, a scientist) then the books of Cipra mentioned in J. O'Rourke's reply
are the best that I know in English.

There is much more of this sort in Russian, where many series like "Popular lectures on
mathematics" are published. These little books are oriented on high school children who have STRONG mtivation to learn something of modern mathematics. Some of these books
are extremelly good. I do not know of any similar project in other languages, and very
few of these books are translated.

II. If a newcomer means a professional mathematician (roughly speaking, a PhD) who is interested in
other areas of mathematics, perhaps very remote from is own, then there are many excellent sources. Some journals which publish such surveys are:
BAMS, Notices AMS, Russian Math Surveys
(translated cover-to-cover), Sugaku expositions, St. Peterburg Journal of Math.,
Expositiones Math., SIAM Reviews, Jahresbericht der DMV, Gazette des Mathématiciens, and probably several other societies journals. I should also mention Seminaire Bourbaki,
though their talks are not necessarily an easy reading:-)
Some excelent introductory surveys can be found in ICM proceedings (all available on the web).

StPeterburg Journal has a special section, called "Easy reading for professionals".

III. Mathematics is so large, that on my opinion, it is impossible for a person, or a small group of people to write an unbiased survey of the whole. For this reason,
Princeton Companion, Penrose's Road to reality, and other similar books, always give somewhat one-sided view. Penrose does not claim giving a complete picture, but Princeton
Companion may be misleading in this respect.

IV. On Andre Henriques answer. I know from experience that people do get invited to write
surveys (in BAMS, Rus. Math. Surveys, StPeterburg J. and probably in other journals).

But there seems to be no centralised repository where one can find all (or most) of them - Mathematics is a bit anarchic. Instead, it would be more profitable for you if you asked for reviews papers about a specific subject you are interested in. Then we (i.e. the MathOverflow community) could refer you to papers that are useful to you.

In other fields of sciences, people get invited to write review articles on given subjects (on which they are experts).
To my knowledge, this does not happen in mathematics, and it is the author's initiative to write a review paper.